Relativizations of the of the Relativizations P =? DNP Question P - - PowerPoint PPT Presentation

Relativizations Relativizations of the

• f the

P =? DNP Question P =? DNP Question for the BSS Model for the BSS Model

Christine Gaßner Christine Gaßner Greifswald Greifswald

Ljubljana 2009 gassnerc@uni-greifswald.de

The The machines machines

Ljubljana 2009 gassnerc@uni-greifswald.de

The The complexity complexity classes classes

Ljubljana 2009 gassnerc@uni-greifswald.de

The The complexity complexity classes classes

Ljubljana 2009 gassnerc@uni-greifswald.de

The The complexity complexity classes classes

Ljubljana 2009 gassnerc@uni-greifswald.de

The The oracle

• racle machines

machines

Ljubljana 2009 gassnerc@uni-greifswald.de

A summary A summary

Ljubljana 2009 gassnerc@uni-greifswald.de

A summary A summary

Similarly to P PQ

Q Q ≠

≠ NP NPQ

Q Q

• cp. CCA 2008

Similarly to P PQ

Q Q ≠

≠ NP NPQ

Q Q

• cp. CCA 2008

Ljubljana 2009 gassnerc@uni-greifswald.de

A summary A summary

• cp. CCC 2009
• cp. CCC 2009

Ljubljana 2009 gassnerc@uni-greifswald.de

A summary A summary

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P PR

R Q Q ≠

≠ DN DNP PR

R Q Q

Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only 0 and 1 as constants 0 and 1 as constants) )

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P PR

R Q Q ≠

≠ DN DNP PR

R Q Q

Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only 0 and 1 as constants 0 and 1 as constants) ) BSS BSS -

• only with 0 and 1
• nly with 0 and 1

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P PR

R Q Q ≠

≠ DN DNP PR

R Q Q

Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only 0 and 1 as constants 0 and 1 as constants) ) BSS BSS -

• only with 0 and 1
• nly with 0 and 1

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P PR

R Q Q ≠

≠ DN DNP PR

R Q Q

Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only 0 and 1 as constants 0 and 1 as constants) ) BSS BSS -

• only with 0 and 1
• nly with 0 and 1

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P PR

R Q Q ≠

≠ DN DNP PR

R Q Q

Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only 0 and 1 as constants 0 and 1 as constants) ) BSS BSS -

• only with 0 and 1
• nly with 0 and 1

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P PR

R Q Q ≠

≠ DN DNP PR

R Q Q

Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only 0 and 1 as constants 0 and 1 as constants) ) BSS BSS -

• only with 0 and 1
• nly with 0 and 1

Important: V V

i i

≠ ∅ ≠ ∅ iff iff N N N

i i W W i i

rejects rejects (0,..., 0) (0,..., 0) ∈ ∈ ℝ ℝ

n n i i

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P PR

R Q Q ≠

≠ DN DNP PR

R Q Q

Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only 0 and 1 as constants 0 and 1 as constants) ) V Vi

i ≠∅

≠∅ can be satisfied can be satisfied BSS BSS -

• only with 0 and 1
• nly with 0 and 1

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P PR

R Q Q ≠

≠ DN DNP PR

R Q Q

Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only 0 and 1 as constants 0 and 1 as constants) )

since since the path of the path of N Ni

i W Wi i traversed by

traversed by (0,..., 0) (0,..., 0) ∈ ∈ ℝ ℝn

ni i

• is uniquely determined

is uniquely determined

• of polynomial length
• f polynomial length

V V

i i

≠∅ ≠∅ can be satisfied can be satisfied

BSS BSS -

• only with 0 and 1
• nly with 0 and 1

Ljubljana 2009 gassnerc@uni-greifswald.de

N Ni

i W Wi i rejects

rejects (0,..., 0) (0,..., 0) ∈ ∈ ℝ ℝ

n ni i

Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only 0 and 1 as constants 0 and 1 as constants) )

N Ni

i W Wi i:

:

Ljubljana 2009 gassnerc@uni-greifswald.de

N Ni

i W Wi i rejects

rejects (0,..., 0) (0,..., 0) ∈ ∈ ℝ ℝ

n ni i

Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only 0 and 1 as constants 0 and 1 as constants) )

Only 0 and 1 as constants Only 0 and 1 as constants encoded by themselves encoded by themselves N Ni

i W Wi i:

:

Ljubljana 2009 gassnerc@uni-greifswald.de

N Ni

i W Wi i rejects

rejects (0,..., 0) (0,..., 0) ∈ ∈ ℝ ℝ

n ni i

Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only 0 and 1 as constants 0 and 1 as constants) )

Only 0 and 1 as constants Only 0 and 1 as constants encoded by themselves encoded by themselves ⇒ ⇒ The polynomials are The polynomials are uniquely determined. uniquely determined. N Ni

i W Wi i:

:

Ljubljana 2009 gassnerc@uni-greifswald.de

N Ni

i W Wi i rejects

rejects (0,..., 0) (0,..., 0) ∈ ∈ ℝ ℝ

n ni i

Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only 0 and 1 as constants 0 and 1 as constants) )

Only 0 and 1 as constants Only 0 and 1 as constants encoded by themselves encoded by themselves ⇒ ⇒ The polynomials are The polynomials are uniquely determined. uniquely determined. ⇒ ⇒ The path is uniquely The path is uniquely determined. determined. N Ni

i W Wi i:

:

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P PR

R Q Q ≠

≠ DN DNP PR

R Q Q

Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only 0 and 1 as constants 0 and 1 as constants) ) BSS BSS -

• only with 0 and 1
• nly with 0 and 1

Ljubljana 2009 gassnerc@uni-greifswald.de

length length ≤ ≤ 2 2n

ni i

⇒∃x ∈ {0, 1}

ni

( ( x x is not queried by is not queried by N Ni

i W Wi i )

) ⇒ ⇒ V Vi

i ≠∅

≠∅

V Vi

i ≠∅

≠∅ iff iff N Ni

i W Wi i rejects

rejects (0,..., 0) (0,..., 0) ∈ ∈ ℝ ℝ

n ni i

Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only 0 and 1 as constants)

(only 0 and 1 as constants)

N Ni

i W Wi i:

:

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P PR

R Q Q ≠

≠ DN DNP PR

R Q Q

Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only 0 and 1 as constants 0 and 1 as constants) )

Important: N N

i i Q Q

≙ ≙ N N

i i W W i i + 1 + 1

≙ ≙ N N

i i W W i i

• n
• n (0,..., 0)

(0,..., 0) ∈ ∈ ℝ ℝ

n n i i

BSS BSS -

• only with 0 and 1
• nly with 0 and 1

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P PR

R Q Q ≠

≠ DN DNP PR

R Q Q

Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only 0 and 1 as constants 0 and 1 as constants) )

⇒ ⇒ N Ni

i W Wi i +1 +1 ≙

≙ N Ni

i Q Q

• n
• n

( , . . . , ) ( , . . . , ) ∈ ∈ ℝ ℝn

ni i

BSS BSS -

• only with 0 and 1
• nly with 0 and 1

Ljubljana 2009 gassnerc@uni-greifswald.de

N Ni

i W Wi i+1 +1 ≙

≙ N Ni

i Q Q on

• n (0,..., 0)

(0,..., 0) ∈ ∈ ℝ ℝ

n ni i

Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only 0 and 1 as constants)

(only 0 and 1 as constants)

s sµ

µ ,

, s sλ

λ ≤

≤ n ni

i+1 +1

⇓ ⇓

≙ ≙ ... ... ∈

∈ W

Wi

i+1 +1 ?

? N Ni

i Q Q:

: ≙ ≙ ... ... ∈

∈ W

Wi

i+1 +1 ?

?

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P PR

R Q Q ≠

≠ DN DNP PR

R Q Q

Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only 0 and 1 as constants 0 and 1 as constants) )

⇒ ⇒N Ni

i W Wi i +1 +1 ≙

≙ N Ni

i W Wi i on

• n (0,..., 0)

(0,..., 0) ∈ ∈ ℝ ℝn

ni i

BSS BSS -

• only with 0 and 1
• nly with 0 and 1

Ljubljana 2009 gassnerc@uni-greifswald.de

N Ni

i W Wi i ≙

≙ N Ni

i W Wi i+1 +1 ≙

≙ N Ni

iQ Q

• n
• n (0,..., 0)

(0,..., 0) ∈ ∈ ℝ ℝ

n ni i

Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only 0 and 1 as constants)

(only 0 and 1 as constants)

x x ∈ ∈ V Vi

i

⇒ ⇒ x x ∈ ∈W Wi

i+1 +1

⇒ ⇒ x x is not queried by is not queried by N Ni

i W Wi i

⇓ ⇓

N Ni

i Q Q:

: ≙ ≙ ... ... ∈

∈ W

Wi

i ?

? ≙ ≙ ... ... ∈

∈ W

Wi

i ?

?

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P PR

R Q Q ≠

≠ DN DNP PR

R Q Q

Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only 0 and 1 as constants)

(only 0 and 1 as constants) BSS BSS -

• only with 0 and 1
• nly with 0 and 1

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P PR

R Q Q ≠

≠ DN DNP PR

R Q Q

Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only 0 and 1 as constants)

(only 0 and 1 as constants) BSS BSS -

• only with 0 and 1
• nly with 0 and 1

N Ni

i W Wi i ≙

≙ N Ni

i Q Q on

• n (0,..., 0)

(0,..., 0) ∈ ∈ ℝ ℝn

ni i

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P Pℝ

ℝ(=) (=) Q Q ≠

≠ DN DNP Pℝ

ℝ(=) (=) Q Q Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only equality tests equality tests) )

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P Pℝ

ℝ(=) (=) Q Q ≠

≠ DN DNP Pℝ

ℝ(=) (=) Q Q Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only equality tests equality tests) )

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P Pℝ

ℝ(=) (=) Q Q ≠

≠ DN DNP Pℝ

ℝ(=) (=) Q Q Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only equality tests equality tests) )

• nly equality tests
• nly equality tests

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P Pℝ

ℝ(=) (=) Q Q ≠

≠ DN DNP Pℝ

ℝ(=) (=) Q Q Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only equality tests equality tests) )

• nly equality tests
• nly equality tests

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P Pℝ

ℝ(=) (=) Q Q ≠

≠ DN DNP Pℝ

ℝ(=) (=) Q Q Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only equality tests equality tests) ) V V

i i

=∅ =∅ can be satisfied although can be satisfied although x is not queried on x is not queried on (0,...,0) (0,...,0)

• nly equality tests
• nly equality tests

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P Pℝ

ℝ(=) (=) Q Q ≠

≠ DN DNP Pℝ

ℝ(=) (=) Q Q Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only equality tests equality tests) ) V V

i i

=∅ =∅ can be satisfied although can be satisfied although x is not queried on x is not queried on (0,...,0) (0,...,0)

• nly equality tests
• nly equality tests

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P Pℝ

ℝ(=) (=) Q Q ≠

≠ DN DNP Pℝ

ℝ(=) (=) Q Q Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only equality tests equality tests) )

• nly equality tests
• nly equality tests

c p . C C C 2 9

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P Pℝ

ℝ(=) (=) Q Q ≠

≠ DN DNP Pℝ

ℝ(=) (=) Q Q Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only equality tests equality tests) )

• nly equality tests
• nly equality tests

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P Pℝ

ℝ(=) (=) Q Q ≠

≠ DN DNP Pℝ

ℝ(=) (=) Q Q Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only equality tests equality tests) )

• nly equality tests
• nly equality tests

The path of Ni

Wi,c1,...,cki is

uniquely determined.

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P Pℝ

ℝ(=) (=) Q Q ≠

≠ DN DNP Pℝ

ℝ(=) (=) Q Q Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only equality tests equality tests) )

• nly equality tests
• nly equality tests

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P Pℝ

ℝ(=) (=) Q Q ≠

≠ DN DNP Pℝ

ℝ(=) (=) Q Q Diagonalization Diagonalization techniques from techniques from Baker, Gill, and Baker, Gill, and Solovay Solovay (only

(only equality tests equality tests) )

• nly equality tests
• nly equality tests

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P Pℝ

ℝ Q Q ≠

≠ DN DNP Pℝ

ℝ Q Q

Problems in the full BSS model Problems in the full BSS model

BSS BSS -

• with order tests

with order tests

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P Pℝ

ℝ Q Q ≠

≠ DN DNP Pℝ

ℝ Q Q

Problems in the full BSS model Problems in the full BSS model

? BSS BSS -

• with order tests

with order tests

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P Pℝ

ℝ Q Q ≠

≠ DN DNP Pℝ

ℝ Q Q

Problems in the full BSS model Problems in the full BSS model

BSS BSS -

• with order tests

with order tests

Characterization Characterization of the algebraic

• f the algebraic

dependence of the constants dependence of the constants c c1

1,...,

,...,c ck

ki i

is is not not sufficient sufficient

!!!

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P Pℝ

ℝ Q Q ≠

≠ DN DNP Pℝ

ℝ Q Q

Problems in the full BSS model Problems in the full BSS model

BSS BSS -

• with order tests

with order tests

⇒ ⇒ We need a new encoding. We need a new encoding. But: If But: If n ni

i will be greater, then the test

will be greater, then the test results are also dependent on the new results are also dependent on the new zeros of the new zeros of the new polynonials polynonials. .

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P Pℝ

ℝ Q Q ≠

≠ DN DNP Pℝ

ℝ Q Q

Problems in the full BSS model Problems in the full BSS model

BSS BSS -

• with order tests

with order tests ?

Ljubljana 2009 gassnerc@uni-greifswald.de

N Ni

i W Wi i, ,c c1 1,..., ,...,c ck ki i rejects

rejects (0,..., 0, (0,..., 0,C Ci

i, , j j)

) ∈ ∈ ℝ ℝ

n ni i ?

?

Using further ideas for the full BSS model Using further ideas for the full BSS model N Ni

i W Wi i, ,c c1 1,..., ,...,c ck ki i∈

∈K

Ki

i, ,j j:

: The The values of the values of the polynomials at polynomials at C

Ci

i, ,j j

are uniquely are uniquely determined by determined by C

Ci

i, ,j j.

.

Ljubljana 2009 gassnerc@uni-greifswald.de

N Ni

i W Wi i, ,c c1 1,..., ,...,c ck ki i rejects

rejects (0,..., 0, (0,..., 0,C Ci

i, , j j)

) ∈ ∈ ℝ ℝ

n ni i ?

?

Using further ideas for the full BSS model Using further ideas for the full BSS model N Ni

i W Wi i, ,c c1 1,..., ,...,c ck ki i∈

∈K

Ki

i, ,j j:

: The The values of the values of the polynomials at polynomials at C

Ci

i, ,j j

are uniquely are uniquely determined by determined by C

Ci

i, ,j j.

. ⇒ ⇒ The path is uniquely The path is uniquely determined. determined.

Ljubljana 2009 gassnerc@uni-greifswald.de

The definition of The definition of C Ci

i, ,j j

BSS BSS -

• with order tests

with order tests

Ljubljana 2009 gassnerc@uni-greifswald.de

BSS BSS -

• with order tests

with order tests

A n e n u m e r a t i

• n

The definition of The definition of C Ci

i, ,j j

Ljubljana 2009 gassnerc@uni-greifswald.de

BSS BSS -

• with order tests

with order tests

All possible coefficients are considered.

The definition of The definition of C Ci

i, ,j j

Ljubljana 2009 gassnerc@uni-greifswald.de

BSS BSS -

• with order tests

with order tests

The definition of The definition of C Ci

i, ,j j

Ljubljana 2009 gassnerc@uni-greifswald.de

The definition of The definition of C Ci

i, ,j j

BSS BSS -

• with order tests

with order tests

Ljubljana 2009 gassnerc@uni-greifswald.de

The definition of The definition of C Ci

i, ,j j

BSS BSS -

• with order tests

with order tests

determines order tests pν,µ(x)≥0 and queries

• n (0,…,0, N)

if N ∈ ℕ, pν,µ qλ,µ∈ℚ[x]

Ljubljana 2009 gassnerc@uni-greifswald.de

The definition of The definition of C Ci

i, ,j j

BSS BSS -

• with order tests

with order tests

Ljubljana 2009 gassnerc@uni-greifswald.de

The definition of The definition of C Ci

i, ,j j

BSS BSS -

• with order tests

with order tests

Ljubljana 2009 gassnerc@uni-greifswald.de

The definition of The definition of C Ci

i, ,j j

BSS BSS -

• with order tests

with order tests

greater than the zeros

Ljubljana 2009 gassnerc@uni-greifswald.de

The definition of The definition of C Ci

i, ,j j

BSS BSS -

• with order tests

with order tests

Ljubljana 2009 gassnerc@uni-greifswald.de

The definition of The definition of C Ci

i, ,j j

BSS BSS -

• with order tests

with order tests

determine order tests pν,µ(x) ≥ 0

• n (0,…, 0, N)

for large N ∈ ℕ

Ljubljana 2009 gassnerc@uni-greifswald.de

The definition of The definition of C Ci

i, ,j j

BSS BSS -

• with order tests

with order tests

determine order tests pν,µ(x) ≥ 0

• n (0,…, 0, N)

for large N ≥ Nchar(…)

Ljubljana 2009 gassnerc@uni-greifswald.de

The definition of The definition of C Ci

i, ,j j

BSS BSS -

• with order tests

with order tests

Ljubljana 2009 gassnerc@uni-greifswald.de

The definition of The definition of C Ci

i, ,j j

BSS BSS -

• with order tests

with order tests

greater than all natural values

Ljubljana 2009 gassnerc@uni-greifswald.de

The definition of The definition of C Ci

i, ,j j

BSS BSS -

• with order tests

with order tests

determine parts of the queries on (0,…, 0, N) for large N ≥ Nchar(…), N ∈ ℕ

Ljubljana 2009 gassnerc@uni-greifswald.de

The definition of The definition of C Ci

i, ,j j

BSS BSS -

• with order tests

with order tests

Ljubljana 2009 gassnerc@uni-greifswald.de

The definition of The definition of C Ci

i, ,j j

BSS BSS -

• with order tests

with order tests

allows to extend the oracle step-by-step by a recursive definition

Ljubljana 2009 gassnerc@uni-greifswald.de

The definition of The definition of C Ci

i, ,j j

BSS BSS -

• with order tests

with order tests

Ljubljana 2009 gassnerc@uni-greifswald.de

The definition of The definition of C Ci

i, ,j j

BSS BSS -

• with order tests

with order tests

Ljubljana 2009 gassnerc@uni-greifswald.de

N Ni

i W Wi i, ,c c1 1,..., ,...,c ck ki i rejects

rejects (0,..., 0, (0,..., 0,C Ci

i, , j j)

) ∈ ∈ ℝ ℝ

n ni i

The The values of the values of the polynomials at polynomials at C

Ci

i, ,j j

are uniquely are uniquely determined by determined by C

Ci

i, ,j j.

. ⇒ ⇒ The path is uniquely The path is uniquely determined. determined. N Ni

i W Wi i, ,c c1 1,..., ,...,c ck ki i∈

∈K

Ki

i, ,j j:

:

Ljubljana 2009 gassnerc@uni-greifswald.de

An oracle An oracle Q Q with with P Pℝ

ℝ Q Q ≠

≠ DN DNP Pℝ

ℝ Q Q

Problems in the full BSS model Problems in the full BSS model

BSS BSS -

• with order tests

with order tests S e e m y p a p e r .

Ljubljana 2009 gassnerc@uni-greifswald.de

A second oracle A second oracle Q Q with with P Pℝ

ℝ Q Q ≠

≠ DN DNP Pℝ

ℝ Q Q

Problems in the full BSS model Problems in the full BSS model

Ljubljana 2009 gassnerc@uni-greifswald.de

A second oracle A second oracle Q Q with with P Pℝ

ℝ Q Q ≠

≠ DN DNP Pℝ

ℝ Q Q

Problems in the full BSS model Problems in the full BSS model

Ljubljana 2009 gassnerc@uni-greifswald.de

yes yes yes yes

A second oracle A second oracle Q Q with with P Pℝ

ℝ Q Q ≠

≠ DN DNP Pℝ

ℝ Q Q

Problems in the full BSS model Problems in the full BSS model

(qµ,1(r),..., qµ,sµ(r)) ∉ Wi ? (qλ,1(r),..., qλ,sλ(r)) ∉ Wi ? Output: 0 ? Input: (0,..., 0, r) ∈ ℝ

n

After a polynomial number of steps

• f a deterministic machine

pν,1(r) ≥ 0 ? pν, j(r) ≥ 0 ?

Ljubljana 2009 gassnerc@uni-greifswald.de

A second oracle A second oracle Q Q with with P Pℝ

ℝ Q Q ≠

≠ DN DNP Pℝ

ℝ Q Q

Problems in the full BSS model Problems in the full BSS model

S e e m y p a p e r .

Ljubljana 2009 gassnerc@uni-greifswald.de

A summary A summary

Ljubljana 2009 gassnerc@uni-greifswald.de

Thank you for your attention! Thank you for your attention!

Christine Christine Gaßner Gaßner

Greifswald Greifswald. .

Thanks also to Thanks also to Volkmar Volkmar Liebscher Liebscher, , the the organizers

• rganizers of the CCA 2009,
• f the CCA 2009,

the referees reading my paper. the referees reading my paper.

Relativizations Relativizations of the P =? DNP Question for the

• f the P =? DNP Question for the

BSS Model BSS Model